We propose some algorithms to solve the system of linens equations arising from the finite difference discretization on sparse grids. For this, we will use the multilevel structure of the sparse grid space or its full...
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We propose some algorithms to solve the system of linens equations arising from the finite difference discretization on sparse grids. For this, we will use the multilevel structure of the sparse grid space or its full grid subspaces, respectively.
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type. The ...
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In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first nontrivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or nonlinear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Muller-Gronbach, B. Niu, and K. Ritter, J. Complexity, 26 (2010), pp. 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.
The efficient and reliable solution of partial differential equations(PDEs) plays an essential role in a very large number of applications in business,engineering and science, ranging from the modelling of financial m...
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The efficient and reliable solution of partial differential equations(PDEs) plays an essential role in a very large number of applications in business,engineering and science, ranging from the modelling of financial markets through to the prediction of complex fluid *** paper presents a discussion of alternative approaches to the fast solution of elliptic and parabolic PDEs based upon the use of parallel, adaptive and multilevel *** adaptivity is essential to ensure that the solution is approximated to different local resolutions across the domain according to its local properties,whilst the multilevel algorithms ensure that the computational time to solve the resulting finite element equations is proportional to the number of unknowns. Applying these techniques efficiently on parallel computer architectures leads to significant practical problems. Difficulties addressed in this paper include how to handle the coarse grid operations efficiently in parallel and the dynamic load-balancing problem that arises when the finite element mesh is adapted.
Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent...
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Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is a restriction of this technique in which mesh entities are only allowed to migrate between subdomains in one direction. The third treats the gain as a potential held and uses the concept of relative gain for selecting appropriate vertices to migrate. The results are compared and seen to produce very high global quality partitions, very rapidly. The results are also compared with another partitioning tool and shown to be of higher quality although taking longer to compute. (C) 2000 Elsevier Science B.V. All rights reserved.
One of the most useful measures of cluster quality is the modularity of the partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of su...
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One of the most useful measures of cluster quality is the modularity of the partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of such edges in a random graph. In this paper, we show that the problem of finding a partition maximizing the modularity of a given graph G can be reduced to a minimum weighted cut (MWC) problem on a complete graph with the same vertices as G. We then show that the resulting minimum cut problem can be efficiently solved by adapting existing graph partitioning techniques. Our algorithm finds clusterings of a comparable quality and is much faster than the existing clustering algorithms.
Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. multilevel methods make more assumptions regarding the struc...
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Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impressive performance of multilevel optimization methods is an empirical observation, and no theoretical explanation has so far been proposed. In order to address this issue, we study the convergence properties of a multilevel method that is motivated by second-order methods. We take the first step toward establishing how the structure of an optimization problem is related to the convergence rate of multilevel algorithms.
multilevel algorithms are a successful class of optimization techniques which addresses the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimization method which...
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multilevel algorithms are a successful class of optimization techniques which addresses the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimization method which refines the partition at each graph level. In this paper we present an enhancement of the technique which uses imbalance to achieve higher quality partitions. We also present a formulation of the Kernighan-Lin partition optimization algorithm which incorporates load-balancing. The resulting algorithm is tested against a different but related state-of-the-art partitioner and shown to provide improved results.
Design and implementation issues that concern the development of a package of parallel algebraic two-level Schwarz preconditioners are discussed. The computations are based on the Parallel Sparse BLAS (PSBLAS) library...
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Design and implementation issues that concern the development of a package of parallel algebraic two-level Schwarz preconditioners are discussed. The computations are based on the Parallel Sparse BLAS (PSBLAS) library. The package implements various versions of Additive Schwarz preconditioners and applies a smoothed aggregation technique to generate a coarse-level correction. The coarse matrix can be either replicated on the processors or distributed among them;the corresponding system is solved by factorization or block Jacobi sweeps, respectively. The design of the package started from a description of the preconditioners in terms of parallel basic Linear Algebra operators, in order to develop software based on standard kernels. Suitable preconditioner data structures were defined to fully exploit the existing PSBLAS functionalities;however, the implementation of the preconditioner required also an extension of the set of basic library kernels. The results of experiments carried out on different test matrices show that the package is competitive in terms of runtime efficiency. (c) 2007 IMACS. Published by Elsevier B.V. All rights reserved.
In this article, we show that multigrid-like algorithms can be obtained by combining space decomposition with time discretization by operator splitting. (C) 2002 Elsevier Science Ltd. All rights reserved.
In this article, we show that multigrid-like algorithms can be obtained by combining space decomposition with time discretization by operator splitting. (C) 2002 Elsevier Science Ltd. All rights reserved.
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